Revisiting Deterministic Diffusion through Reverse Transition Kernels

Deterministic diffusion samplers replace stochastic reverse dynamics with an iterative reconstruction map, but the conditions under which this iteration is stable remain poorly understood. We study Directly Denoising Diffusion Models (DDDMs), which refine an estimate of the clean sample by a fixed-point iteration at each noise level. Through a reverse transition kernel formulation, each reverse-time step defines a residual energy, and Tweedie's identity yields a certificate of local strong convexity for this energy—a sufficient condition for stable refinement. We audit this certificate at scale on trained models and report three findings. First, the derivative that governs convergence is taken with respect to the sampler's own running estimate, whereas Tweedie's identity concerns derivatives with respect to the noisy observation; on trained networks these are measurably different operators, and only the latter exhibits the symmetry Tweedie predicts. Second, the curvature term in the certificate can be computed exactly rather than bounded, tightening it by a median factor of $3$–$33$. Third, across $960$ configurations spanning variance-preserving models under linear and cosine schedules and variance-exploding models, the energy is strongly convex more than twice as often as even the exact certificate detects ($69$% versus $29$%, with the worst-case bound certifying almost none); we trace the remaining conservatism to a single geometric assumption—that the certificate's two terms act along a common worst-case direction, where in measurement they are nearly orthogonal. A one-dimensional system with closed-form posterior statistics reproduces the predicted contraction rate to within $1.7$% and shows that the certificate, not the sampler, loses resolution at low noise: where certification fails, the iteration itself still converges.